Semigroup Forum

, Volume 90, Issue 2, pp 532–544 | Cite as

Partial transformation monoids preserving a uniform partition

  • Serena Cicalò
  • Vítor H. Fernandes
  • Csaba Schneider
RESEARCH ARTICLE
  • 131 Downloads

Abstract

The objective of this paper is to study the monoid of all partial transformations of a finite set that preserve a uniform partition. In addition to proving that this monoid is a quotient of a wreath product with respect to a congruence relation, we show that it is generated by 5 generators, we compute its order and determine a presentation on a minimal generating set.

Keywords

Transformation monoids Uniform partitions Ranks  Presentations Partial transformations Wreath products  

References

  1. 1.
    Aĭzenštat, AJa: Defining relations of finite symmetric semigroups. Mat. Sb. N.S. 45(87), 261–280 (1958)MathSciNetGoogle Scholar
  2. 2.
    Araújo, I.M.: Presentations for semigroup constructions and related computational methods. PhD thesis, University of St Andrews (2001)Google Scholar
  3. 3.
    Araújo, J., Schneider, Cs: The rank of the endomorphism monoid of a uniform partition. Semigroup Forum 78(3), 498–510 (2009)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    East, J.: A presentation of the singular part of the symmetric inverse monoid. Commun. Algebra 34(5), 1671–1689 (2006)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    East, J.: A presentation for the singular part of the full transformation semigroup. Semigroup Forum 81(2), 357–379 (2010)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    East, J.: Presentations for singular subsemigroups of the partial transformation semigroup. Int. J. Algebra Comput. 20(1), 1–25 (2010)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    East, J.: Generators and relations for partition monoids and algebras. J. Algebra 339, 1–26 (2011)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    East, J.: On the singular part of the partition monoid. Int. J. Algebra Comput. 21(1–2), 147–178 (2011)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Easdown, D., East, J., FitzGerald, D.G.: A presentation of the dual symmetric inverse monoid. Int. J. Algebra Comput. 18(2), 357–374 (2008)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Fernandes, V.H.: Presentations for some monoids of partial transformations on a finite chain: a survey. In: Gomes, G.M.S., Pin, J.-É., Silva, P. V. (eds.). Semigroups. Algorithms, Automata and Languages (Coimbra, 2001), pp. 363–378. World Sci. Publ, River Edge, NJ (2002)Google Scholar
  11. 11.
    Fernandes, V.H., Quinteiro, T.M.: On the monoids of transformations that preserve the order and a uniform partition. Commun. Algebra 39(8), 2798–2815 (2011)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Fernandes, V.H., Quinteiro, T.M.: The cardinal of various monoids of transformations that preserve a uniform partition. Bull. Malays. Math. Sci. Soc. (2) 35(4), 885–896 (2012)MATHMathSciNetGoogle Scholar
  13. 13.
    Fernandes, V.H., Quinteiro, T.M.: On the ranks of certain monoids of transformations that preserve a uniform partition. Commun. Algebra 42(2), 615–636 (2014)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Ganyushkin, O., Mazorchuk, V.: Classical Finite Transformation Semigroups, An introduction, volume 9 of Algebra and Applications. Springer London Ltd, London (2009)Google Scholar
  15. 15.
    Huisheng, P., Dingyu, Z.: Green’s equivalences on semigroups of transformations preserving order and an equivalence relation. Semigroup Forum 71(2), 241–251 (2005)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Huisheng, P.: On the rank of the semigroup \(T_E(X)\). Semigroup Forum 70(1), 107–117 (2005)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Lavers, T.G.: Presentations of general products of monoids. J. Algebra 204(2), 733–741 (1998)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Moore, E.H.: Concerning the abstract groups of order \(k!\) and \(\frac{1}{2} k!\) holohedrically isomorphic with the symmetric and the alternating substitution-groups on \(k\) letters. Proc. Lond. Math. Soc. 28(1), 357–367 (1896)CrossRefGoogle Scholar
  19. 19.
    Popova, L.M.: Defining relations of a semigroup of partial endomorphisms of a finite linearly ordered set. Leningrad. Gos. Ped. Inst. Učen. Zap. 238, 78–88 (1962)MathSciNetGoogle Scholar
  20. 20.
    Rhodes, J., Steinberg, B.: The \(q\)-theory of Finite Semigroups. Springer Monographs in Mathematics. Springer, New York (2009)Google Scholar
  21. 21.
    Ruškuc, N.: Semigroup presentations. PhD thesis, University of St Andrews (1995)Google Scholar
  22. 22.
    Sun, L., Pei, H., Cheng, Z.: Regularity and Green’s relations for semigroups of transformations preserving orientation and an equivalence. Semigroup Forum 74(3), 473–486 (2007)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Serena Cicalò
    • 1
  • Vítor H. Fernandes
    • 2
    • 3
  • Csaba Schneider
    • 4
  1. 1.Dipartimento di Matematica e InformaticaUniversità degli Studi di CagliariCagliariItaly
  2. 2.Departamento de Matemática, Faculdade de Ciências e TecnologiaUniversidade Nova de Lisboa, Monte da CaparicaCaparicaPortugal
  3. 3.Centro de Álgebra da Universidade de LisboaLisbonPortugal
  4. 4.Departamento de Matemática, Instituto de Ciências ExatasUniversidade Federal de Minas GeraisBelo HorizonteBrazil

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